What does Sin, Cos, Tan actually mean? Trigonometry explained for Beginners!

so sine cosine and tangent right what are these words what part of speech are they that's what I'm gonna start with are they a verb are they an adjective are they a noun well yeah they're a noun so what is it now right and now is is usually things things like let's say glasses right I've got a pair of glasses here you know they look like this you can put them on pretty cool stuff right what else is a noun let's see what have I got in my pocket I've got a phone here in my pocket a smartphone right you can see that's what a smartphone looks like you might be asking well sine cosine and tangent they're nouns so they must be things so what do they look like so the thing is they don't look like anything because in fact they're not a tangible object so you can have a now that's not a tangible object you might be saying well I want you to think of the word price okay so what's the definition of price so you could say it's the amount of money you need to pay to acquire a product so if I were to say show me what a price looks like you're gonna be a bit stuck because you could show me a picture of a Christ but that's as a matter of fact a price tag that's not price okay so price is a noun but it's it's kind of a concept okay it's kind of a relationship between a product and the Maui have to pay now in mathletics we call these words functions a function is when you have an input and an output okay so what does it actually mean so in terms of the word price okay the input is a product at a shop so you go to a shop any product at that shop that's for sale can be the input to the price function and then the output is simply a number right with a unit for the currency of whatever country you're in I'm currently in Japan so the unit would be yet right so I could go to a supermarket find an apple with an apple into the price function and then the output would be let's say 78 yen so the reason why I brought up the word price is because sine cosine and tangent are used in the exact same way so if you couldn't tell already these words sine cosine and tangent they're nouns they're not tangible and they're functions so you have an input and you have an output except in the input you're no longer putting in a physical product data store you're putting in something else you're putting in an angle right so what is an angle again right an angle with I looks like this right that's about 30 degrees that's about 90 degrees also known as a right angle and it can go all the way to 360 degrees which is a full rotation right that is an angle so you can put any size of an angle between 0 and 360 degrees into the sine cosine or tangent function as you would with putting in any product at a store into the price function so let's say we pick a random angle let's say 30 degrees okay and then you can put it into let's say the sine function so you put 30 degrees into the sine function so that becomes a sine of 30 is 0.5 right we'll get to why that 0.5 in a little bit but that's the same sentence structure as saying the price of Apple is 78 yen all right the sine of 30 is 0.5 so now we're going to cover the mathematical notation of functions so in English we would say for instance the price of Apple is 78 yen okay so I'm just going to write most use red color for the hospital that's my recommend there's okay is the Apple is the input to the price function seventy-eight yes right so that is the function written down in English all right so we can see this part here this is the name of the function and then this here this is the input and then this here is the outfit and in maths we write this in a slightly different way so we start off the same way with naming the function so we would say price but now instead of putting off we use brackets okay so we're gonna open our bracket here this is the same thing as saying of and then we're going to close the brackets here now inside the brackets goes our input so we're gonna put Apple because that is our input so Apple in the input and then instead of saying is we write the equal sign like so which means the same thing and then we have the output at the end 78 yeah okay so that is the price function written in mathematical notation so we can actually apply the same thing for sine cosine and tangent okay the ambulance is a bit and loud isn't it okay let's let's get rid of this okay so let's use the example from earlier okay we had of and then we infer which I will write in red a sine of 30 degrees yes okay and now they out helicopter so much noise okay a helicopter is gone so we have the sine of 30 degrees is the output which is 0.5 so again we have the name of the function and then the input and then the out them and in mathematical notation we would write this sentence like this so we would start with sign calling the function except we usually abbreviate sign and take off the e on the end makes life a little bit easier I suppose and then the brackets okay which means all and inside the brackets goes our input which in this case is 30 degrees so we have sine 30 degrees and we tend not to book the degree sign just because it's assumed that the number we're putting that in there is already in degrees and then we have is written as an equal sign and then we have the output which is unchanged of course so this here is this sentence put into mathematical notation and this is the way you're going to see sine cosine and tangent used so you can do the same thing for cosine cosine with empty brackets they're equals and then the output and then we can have tangent with open brackets with an equal sign there so we can pick some input for cosine we can say let's hear 90 and then the output is going to be zero and I'll get to that later why all these values are these values or we can have tan of 45 degrees and this is going to output 1 so we would read these sentences this way so we say sine of 30 is 0.5 and cos 90 or cos of 90 is 0 tan of 45 is 1 or we can skip the off and we just simply say sine 30 is 0.5 cos 90 is 0 tan 45 is 1 so that's the way you're gonna hear these functions be used ok so now I want to introduce you to an open function meaning that basically we haven't picked an input for the function yet so you would think ok you might write the function like this the sine function with open brackets because we don't know what we're putting in there yet which makes sense but in maths we like to put in a what is known as a variable we want to put in a letter in there which can be anything we tend to pick something like X and that just means the same thing as having an empty space for the him foot and X we can say can equal any value between 0 degrees and 360 degrees which means we can pick any angle between 0 degrees and 360 degrees to be x2 then put into the sine function and oftentimes in math textbooks you will probably see instead of X you might see something that looks like this and this is known as beta it's a Greek letter and we tend to use theta because theta is often associated with angles and of course the input to a sine cosine tangent function is an angle so we tend to use a variable theta but don't be don't be scared by this year this is just the same thing as saying sine of X or sine of any letter really we can even have something like sine of J okay so before we jump into what the sine cosine tangent function does I need to introduce you to this concept called a unit circle now I'm just looking for a stick that we can draw on the ground with let's have a look I think this should work here okay so I've got my stick here I'm gonna draw the X&Y coordinate plane which looks like this you've got a straight ball I just snapped my stick so we've got the vertical straight line which is the y-axis and we've got the horizontal straight line which is the x-axis so it looks something like that okay this direction here is positive x and up there Y becomes positive Y he comes negative as it comes down X becomes negative as it goes to the left so let's say this point here is one one unit away from the sensor and so is this point here this point okay so now the unit circle is basically a circle which has the center in the center of the coordinate axis and it has a radius of one so the circle I'm gonna try and draw is going to go around here go through all of the points we just drew yeah so this here this is the unit circle okay every point on the circle is exactly one unit away so this length here is one let's pick a point here this point here is one unit away from the center this point here is one unit away from the center okay so to summarize just basically a circle where's radius of one unit which is in the center of the X and y axis okay so we've made it now to the main meat of the video this is where you're gonna understand get a visual understanding of sine cosine tangent so here we have the units so instead of explaining I'm just gonna show you this magnet here when it's there it's zero degrees we're gonna move it up up here when it's there it's 90 degrees okay when it's here it's 45 degrees if you couldn't tell we're talking about the angle okay when we draw the point from the magnet to the center we're talking about this angle here this angle is 45 so when this is here that red line is on top of the blue line so the angle is zero the angle between the red and blue line is zero degrees so when it's up here the red line is here okay so we're talking about the angle between this line and this line which is of course a right angle so when the magnet is there that is 90 degrees so let's move it beyond 90 degrees let's move it to here now this is 180 degrees let's move it again 90 degrees over here we add 90 to that so now this is 270 degrees okay let's put it in between here so that it's exactly between 90 and 180 we're talking about this angle here okay this is going to be 135 degrees so let's move this magnet over here so that we moved it an extra 30 degrees beyond 180 degrees so now we're talking about this angle all the way around to here which is going to be 210 degrees okay so I hope you get the idea we're starting at 0 degrees and we're going anti-clockwise like so and a 4 rotation is 360 degrees so when we get back here is 360 degrees and we're back to 0 degrees at this point we could say this is at 360 degrees or 0 degrees it doesn't matter here 45 degrees 90 degrees 135 degrees we can put it somewhere in between we can say this is probably about 70 degrees what we're interested in wherever this magnet is that okay let's say it's here what we're interested in is the angle the this line from the center to the magnet oh that's bit wonky this line from the center to the magnet makes with the positive horizontal line this line here we're interested in that angle so we could say that is going to be 120 degrees here we say that that angle is about 120 degrees so we can say wherever this magnet is here here that magnet is going to have an x and y coordinate the x-coordinate of course being how much is shifted horizontally and the y-coordinate is how much it's shifted vertically okay so I'm just going to write that down X is horizontal Y is vertical shift okay so remember from earlier this is zero degrees so when this magnet is at zero degrees okay we can say the x coordinate which is its horizontal shift is 1 because remember to the right is positive and we have Y which is its vertical shift is zero effectively its height and let's move this bad boy to it's 90 degrees position so when it's at 90 degrees okay we have the X which is this horizontal shift is now at zero because it's on the horizontal because it's on the y axis now and we have y the y coordinate which is its vertical shift is positive 1 because it's one went up by one unit so let's move it so the 180 degree position now we have the x-coordinate is actually negative one because it's shifted to left the left and down is always negative and remember it's a unit circle so the radius of the circle is one so it's negative one and Y is zero because it's on the y equals zero line now if we move this here this is 270° we have X is zero and we have Y is negative 1 because it's went down 1 so vertically has shifted down by one unit now we can do things in between as well for instance here you might say what's the coordinate of that point when the angle there is 45 degrees so we have 45 degrees X is and how will you work that out well I can work it out with a calculator okay and I have the value which is zero point seven zero seven approximately it keeps on going and y equals also 0.707 so you can see why they're the same value because when it's at 45 degrees exactly in between this point and this point okay it's x value which is its horizontal shift this length here is each - why which is its vertical shift so the red line and the black line are equal in length so let's take another point let's say here okay 210 so basically we're interested in this angle all the way across here let's say that's 210 degrees now the x-coordinate would it be negative or positive it would be negative because this horizontally shifted to the left of center so basically we're talking about this here and that again I need to check with my calculator is going to be cosine of 210 degrees which is- 0.866 and we have the y-value which is its vertical shift and that's also going down from the zero line so we can say that that's probably gonna be negative as well so what's that gonna be that's gonna be use my calculator sienten that's going to be negative zero point five so we're saying that this thing has gone down by negative zero point five units so this should be half and then this here has went to the left by zero point eight six six units okay so why am i talking about the horizontal shift and the vertical shift as this magnet moves around this unit circle okay what does that have to do with sine and cosine I thought we're talking about trigonometry triangles actually this is sine and cosine okay the horizontal shift at each angle on the unit circle is the cosine of that angle and the vertical shift the y-coordinate for any point on the unit circle is the sine of that angle okay so I'm just gonna give you a few seconds to to digest what I just said there okay so let me let me explain a little bit further so let's say this magnet is here so it is at the point where it's 45 degrees so this angle here is 45 degrees and we said the x coordinate is zero point seven oh seven and the y coordinate is zero point seven oh seven and in this case so we have the horizontal shift is zero point seven oh seven vertical shift is zero point seven oh seven so if we input 45 let me just erase we sign and then the input 45 is going to output this value here sign is the vertical shift okay so y equals verb to shift equals sign and the X which we said equals the horizontal shift is cosine so sine of 45 is 0.7 oh seven and the cosine of 45 is also 0.7 or seven because it's the same thing sine of 45 is the same thing as saying the y-coordinate for the vertical shift when this magnet is at the position 45 degrees on the unit circle so if we want to find sine of any angle let's say 90 degrees we place a dot signified by a magnet at the 90 degrees position at the input so that's that is 90 degrees and then sine is the vertical shift and it's vertically shifted upwards by one unit so sine of 90 is in fact one so where would sign so let's move it to here where the angle is 180 180 is zero because it's vertical shift at this point is zero and we can actually check this with our calculator that's true so we have my calculator here okay let's put sine and let's place 90 degrees into sine and we get one and we can put sine of 180 degrees which we said would be zero and it is indeed zero so where would sign be negative one well it would be here right where it's shifted negative one units also you could say it's shifted one unit down so this year the angle here would be around there all the way to the Riesling right sign 270 is we guessed it's going to be negative one so let's just try it in our calculator and see if it's true okay sign of 270 is indeed negative 1 and we can do any end ok we can find the vertical displacement of any point on the circumference any angle so we can say I want to eat here okay where it's 70 degrees so I want to find the vertical displacement this length here when it's shifted 70 degrees that way so we can say just sine of 70 equals and then that is going to give us the vertical component so sine of 70 degrees is going to give us zero point nine three nine six continuing so we have sine of 70 is zero point okay so that is basically all that sine is doing you put the input in and the output is basically the vertical shift and cosine on the other hand is the same thing but it's the horizontal shift so if we draw in our small unit circle we have cosine which is the horizontal of any angle that we have we want to take the cosine of zero degrees so zero degrees we place this magnet at the zero degrees position and we look at the horizontal shift those shifted one units of the right so it's gonna be one let's try cosine of here 270 degrees and that's gonna be zero right because it's vertically shifted but it's horizontally it hasn't shifted at all so let's try another one let's try this one here cosine of 180 degrees so it's negative 1 because it's shifted one unit to the left now we can do in-betweens as long we can try this here 30 degrees so we're interested in this angle here cosine of 30 degrees and we want to find its horizontal shift so we're talking about this length here that length is going to be cosine 30 zero point eight six zero zero point eight six zero point eight six six okay so to explain the tangent function I'm going to be using my laptop so here we have the unit circle if we zoom out we can see the vertical y-axis and the horizontal x-axis and the unit circle right in the center of the two axes with a radius of one unit so we can rotate this radius any amount of degrees so let's rotate it to around there now we can let this angle there be X so then the tangent function tan of X let's leave a space and we're going to copy this into the tangent function so tan of x equals the slope of the red line now the slope also known as a gradient is given by the formula rise over run and in this graph the rise is this length here and then the run of this point here is like so so from our previous section of the video we know that the rise is the vertical shift which is the y coordinate which is sine of X and we know that the horizontal shift this line here is because of the angle and we said that the tangent of an angle is the slope and the slope is its rise over run we have a very useful formula which is that tangent of an angle is equal to sine over cosine so let's just take either way and what would the tangent of the angle 0 be so when the redline the radius is at zero degrees there is no slope so the tangent of 0 degrees is zero because rise over run there is no rise and the run is 1 so it's 0 over 1 which is 0 what about when it's at 90 degrees tangent of 90 gives us an error why because let's do the rise over run the rise also known as sine of X sine of 90 is 1 is shifted one unit vertically upwards but horizontally cosine of X is 0 so we have 1 divided by 0 and you can't divide numbers by 0 it gives us an error it's undefined so try tangent of 90 in any calculator it will give you some sort of error so another interesting one to look at is tangent of 45 which is right around here tangent of 45 have a guess what it is it says rise over run this is its rise this is its run sine x over cos x sine 45 over cos 45 and gives us 1 because in fact sine 45 equals cosine 45 in other words the length of this line here is the same as the length of this line here when this angle here is 45 Wow that's pretty big ok so now I want to show you how everything connects so we have this is sine of this angle and then cosine of this angle is this line is horizontal shape I'm going to make cosine blue and I'm going to make sine nice green color what we can do is we can shift this blue line all the way down so now what do we have I'm just gonna highlight what we have and copy it over here see we have a triangle and what kind of triangle is it it is a right-angled triangle which is the basis of trigonometry so now that's all I've got time for today and I understand obviously this video isn't going to get you a hundred percent in your next trigonometry exam but I hope that gave you a visual understanding instead of just memorizing that sine is opposite over hypotenuse cosine is adjacent over that and I will be posting more videos about trigonometry as well as other mathematical concepts on my channel so if you would like to subscribe that would mean a lot to me but this spin Jake thank you very much for watching and I'll see you in the next video

Channel: Jay McCaughrean

Views: 34,199

Rating: 4.8963485 out of 5

Keywords: trig, triganometry, trig functions, triganometric functions, unit circle, circle triganometry, circle, sine, cosine, tangent, sin, cos, tan, sohcahtoa

Id: Pn1-DLihSh4

Length: 35min 57sec (2157 seconds)

Published: Mon Dec 23 2019